Understanding Euler's Formula: Degrees or Radians?

Euler's formula, a fundamental concept in mathematics, elegantly combines exponential, trigonometric, and complex number concepts. The formula is stated as

eis cos(s) i sin(s)

where s is the angle. A common point of confusion is whether this angle must be in radians or can be in degrees. This article aims to clarify this and provides insights into Euler's formula's application.

Angle in Radians

The angle in Euler's formula must typically be in radians for it to work correctly. This is because the trigonometric functions cos and sin are fundamentally defined in terms of radians in mathematical contexts. To convert an angle from degrees to radians, the formula is:

Radians Degrees × (π / 180)

For example, if you have an angle of 90 degrees, converting it to radians would involve the following calculation:

Radians 90 × (π / 180) π / 2 ≈ 1.5708

Therefore, the Euler's formula for an angle of 90 degrees would be:

ei(π/2) cos(π/2) i sin(π/2)

This conversion is necessary to ensure consistency and accuracy in mathematical computations.

Trigonometric Functions and Radians

In a general context of mathematical formulae, angles are often measured in radians. This is because the cos and sin functions are derived using radians in calculus, leading to specific properties like the derivatives of these functions:

dcos(x) / dx -sin(x) dsin(x) / dx cos(x)

These properties are only true when the angle is in radians. For instance, consider the identity:

sin(x y) sin(x) cos(y) cos(x) sin(y)

This identity holds in radians but not in degrees, which is another reason why radians are preferred.

The Importance of Consistency

It is essential to be consistent with the choice of angle measurement, whether it is in radians or degrees. For example, if you are using an angle in degrees within the formula, you need to ensure that all subsequent trigonometric functions also use degrees. This ensures that the mathematical properties and identities remain valid.

The Calculator Factor

Moving on to practical applications, pocket calculators often offer a degree mode for trigonometric functions. In this mode, the calculator automatically converts the input angle to radians for the calculation. However, when dealing with the exponential function, no such automatic conversion exists. This is a crucial consideration for students and professionals using calculators for complex computations.

Taylor Series and Radians

The ties between Euler's formula and radians extend to the Taylor series expansions of the sine and cosine functions. These series require the angle to be in radians for the terms to align correctly. The Taylor series for sine and cosine are:

Taylor Series for Sine:

sin(x) x - x3/3! x5/5! - ...

Taylor Series for Cosine:

cos(x) 1 - x2/2! x4/4! - ...

When using these series, the angle x must be in radians. Radians are an absolute measure of angle, defined as the ratio of the arc length to the radius. Degrees, on the other hand, are a relative measure, arising from a cultural decision to divide a circle into 360 equal parts.

Given the importance of radians in mathematical properties and series expansions, it is imperative to use radians consistently in Euler's formula to maintain accuracy and avoid potential errors in calculations.